Functions — Paper 4 · IGCSE Mathematics - International 0607

12 f(x) = 4x + 2 g(x) = 5 − 2x h(x) = x2 − 3 (a) Find g(−3). … [1] (b) Find f(h(2)). … [2] (c) Find x when f(x) = −10. x = … [2] (d) Write down the range of h(x). … [1] (e) Find f−1(x). f−1(x) = … [2] (f) k(x) = 10 − 4x Describe fully the single transformation that maps the graph of y = g(x) onto the graph of y = k(x). … … [3] J N 2 (g) The graph of y = h(x) is translated by the vector KK OO . 0 L P Find the equation of the graph of the image. Write your answer in the form y = ax2 + bx + c. y = … [3]

14 marks

Mark scheme: 12(a) 11 1 12(b) 6 2 B1 for h(2) = 1 soi or 4(x2 – 3) + 2 or better 12(c) –3 2 M1 for 4x = –10 – 2 12(d) h(x) ⩾ –3 1 Allow y ⩾ –3 12(e) x − 2 2 y 2 oe final answer M1 for y – 2 = 4x or x = 4y + 2 or = x + 4 4 4 12(f) Stretch 3 B1 for each x-axis invariant [Scale factor] 2 OR Reflection M1 y = −2.70 + 6.75 A2 OR Rotation M1 (2.5, 0) A1 167 (167.47) or 12.5 (12.53) A1 clockwise 12(g) [y =] x² – 4x + 1 3 M2 for y = (x – 2)² – 3 or M1 for x – 2 seen in a quadratic If 0 scored, SC1 for y = x² + 4x + 1

This question in 0607/42 May/June 2017

6 y 6 x 0 10 f(x) = x - 5 log x (a) On the diagram, sketch the graph of y = f(x) for 0 1 x G 10 . [2] (b) Find the co-ordinates of the local minimum point. ( … , … ) [2] (c) Find the range of f(x) for the domain 1 G x G 5 . … [2] (d) Solve the equation f(x) = 2. x = … or x = … [2] (e) Solve the inequality f(x) 1 2. … [1] (f) (i) Find f(0.001), f(0.000 01) and f(0.000 000 1). f(0.001) = … , f(0.000 01) = … , f(0.000 000 1) = … [1] (ii) Complete the statement. The y-axis is … to the graph of y = f(x). [1]

11 marks

Mark scheme: 6(a) Correct sketch 2 B1 for correct shape 6666 5555 4444 3333 2222 1111 0000 0000 2222 4444 6666 8888 10101010 6(b) (2.17, 0.488) or (2.171…, 0.4877…) 2 B1 for each 6(c) 0.488 - f ( x ) - 1.51 2 FT their 0.488 or 0.4877... - f ( x ) - 1.505... B1 for 0.488 - f ( x ) oe or f ( x ) - 1.51 oe 6(d) 0.502 or 0.5015… 2 B1 for each 5.83 or 5.827… 6(e) 0.502 < x < 5.83 1 FT their (d) or 0.5015... < x < 5.827... 6(f)(i) 15.[0] or 15.00… 1 25.[0] or 25.00… 35. [0] or 35.00… 6(f)(ii) [an] asymptote oe 1

This question in 0607/43 May/June 2017

8 f ()x = x 2 + 1 g ()x = 3 + 2x h (x) = , x ! - 1 x + 1 (a) Find f (- 3) . … [1] (b) Find the value of g(h(1)). … [2] (c) Simplify f(g(x)) + f(x). … [3] (d) Find h -1 ()x . h -1 ()x = … [3] (e) Solve. (i) g(x) = 1 x = … [2] (ii) g -1 ()x = 1 x = … [1]

12 marks

Mark scheme: 8(a) 10 1 8(b) 4 2 1 M1 for [h(1) =] or for 2  1  [gh(x) = ] 3 + 2    x + 1  8(c) 5 x 2 + 12 x + 11 3 M1 for (3 + 2 x ) 2 + 1 + x 2 + 1 B1 for 9 + 6 x + 6 x + 4 x 2 or better for (3 + 2 x ) 2 8(d) 1 1−x 3 M1 correct first step − 1 or oe final answer M1 correct second step x x 8(e)(i) – 1 2 M1 for 3 + 2x = 1 8(e)(ii) 5 1

This question in 0607/43 May/June 2017

4 y 5 x –3 0 4 –5 x f (x) = 2 (x - x - 2) (a) On the diagram, sketch the graph of y = f (x) for values of x from –3 to 4. [3] (b) Find the two values of x for which f(x) does not exist. … , … [2] (c) When k ! 0 , write down the number of solutions to the equation f (x) = k . … [1] (d) g (x) = 2 -x + 1 (i) On the diagram, sketch the graph of y = g (x) for - 2 G x G 4 . [2] (ii) Write down the equation of the asymptote to the graph of y = g (x) . … [1] (e) Solve the equation f (x) = g (x) . x = … or x = … [2]

11 marks

Mark scheme: 4(a) Correct sketch 3 B1 for correct middle branch B1 for correct left hand branch 4444 B1 for correct right hand branch 2222 3333 -2-2-2-2 -1-1-1-1 0000 0000 1111 2222 3333 4444 -2-2-2-2 -4-4-4-4 4(b) – 1 2 B1 for each 2 4(c) 2 1 4(d)(i) Correct sketch 2 Must intersect y-axis and be above x-axis 4444 B1 for decreasing exponential graph 2222 3333 -2-2-2-2 -1-1-1-1 0000 0000 1111 2222 3333 4444 -2-2-2-2 -4-4-4-4 4(d)(ii) y = 1 oe 1 4(e) – 0.892 or – 0.8919 to – 0.892[0] 2 B1 for each 2.62 or 2.622 to 2.623

This question in 0607/42 Oct/Nov 2017

11 f (x) = 2 x + 1 g ( x) = x 2 + 1 h (x) = log x (a) (i) Find the value of f(4.5). … [1] (ii) Find the value of h(f(4.5)). … [1] (b) Find f -1 (x) . f -1 (x ) = … [2] (c) Find g(f(x)) in the form ax 2 + bx + c . … [3] (d) p (x) = x 2 - 1 Find the single transformation that maps the graph of y = g(x) onto the graph of y = p(x). … [2] (e) Solve the equation h -1 (x) = 1000 . x = … [1]

10 marks

Mark scheme: 11(a)(i) 10 1 11(a)(ii) 1 1 11(b) x − 1 2 M1 for y – 1 = 2x or x = 2y + 1 or oe 2 y 1 = x + 2 2 11(c) 4 x 2 + 4 x + 2 final answer 3 M1 for (2 x + 1) 2 + 1 B1 for [(2 x + 1) 2 = ] 4 x 2 + 2 x + 2 x + 1 oe 11(d) Translation 2 B1 for each  0     − 2  11(e) 3 1

This question in 0607/42 Oct/Nov 2017

1 f (x) = 3x - 2 g (x) = , x =Y 0 h ( x) = (x + 2) 2 x (a) Find (i) f(4), … [1] (ii) gf(4). … [1] (b) Find g(g(5)). … [2] (c) Solve f(h(x)) = 10. x = … or x = … [3] (d) Find g(h(f(x))) in terms of x. … [2]

9 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 10 1 1(a)(ii) 0.1 1 1(b) 5 2 M1 for g(5) = 0.2 oe 1(c) 0 and –4 nfww 3 M1 for h(x) = 4 or 3( x + 2) 2 − 2[ = 10] B1 for ( x + 2) 2 = 4 oe or 3 x 2 + 12 x = 0 oe 1(d) 1 1 2 M1 for (3 x − 2 + 2) 2 or oe final answer 9x 2 ( 3x ) 2

This question in 0607/43 Oct/Nov 2017

11 f ()x = 10 x g ()x = 2x - 1 (a) Find the value of g(3). … [1] (b) Find the range of f (x) for the domain {-1, 0, 1, 2}. { … } [2] (c) Find x when g ()x = 12 . x = … [2] 2 (d) The graph of y = g (x) is translated by the vector onto the graph of h(x). e3o Find h(x). Give your answer in its simplest form. h(x) = … [3] (e) Find f -1 ()x . f -1 ()x = … [2] (f) tan (g (x)) = 1 and 0° G x G 180° . Find the two values of x. x = … or x = … [3]

13 marks

Mark scheme: 11(a) 5 1 11(b) 0.1oe , 1, 10, 100 2 B1 for 3 correct or all correct seen and spoilt. 11(c) 6.5 oe 2 M1 for 2x – 1 = 12 11(d) 2x – 2 or 2(x – 1) 3 B2 for correct unsimplified answer OR M1 for substituting x − 2 for x M1 for adding 3 to a function in x oe OR M1 for y = 2x + c (c ≠ – 1) leading to answer with gradient 2 M1 for substituting coords of valid point into y = 2x + c 11(e) log x 2 M1 for log y = x or x = 10y 11(f) 23, 113 3 B2 for 23 or B1 for [g(x) =] 45 soi

This question in 0607/41 May/June 2018

11 f ()x = 2x - 7 g ()x = x h ()x = , x =Y 0 x (a) (i) Find f (3). … [1] (ii) Solve f ()x = 1. x = … [2] (b) Find f - 1 ()x . f - 1 ()x = … [2] (c) (i) Find f (g (x)) in terms of x. … [1] (ii) Solve f (g (x)) = 5 . x = … [3] (d) (i) Find h (g (f (x))) in terms of x. … [2] (ii) Find an inequality in terms of x for which h (g (f (x))) exists. … [2]

13 marks

Mark scheme: 11(a)(i) –1 1 11(a)(ii) 4 2 M1 for 2x – 7 = 1 or better 11(b) x + 7 2 y 7 oe M1 for y + 7 = 2x or = x − or x = 2y – 7 2 2 2 Allow f(x) for y 11(c)(i) 2 x − 7 final answer 1 1 Allow x 2 for x 11(c)(ii) 36 3 5 + 7 M2 for x = or better e.g. x = 6 2 or M1 for 2 x − 7 = 5 or their (c)(i) = 5 11(d)(i) 1 2 M1 for 2 x − 7 oe final answer 2 x − 7 1 If 0 scored SC1 for answer their(c)(i) 11(d)(ii) x > 3.5 2 M1 for 2 x − 7 > 0

This question in 0607/42 May/June 2018

12 f(x) = 2x + 1 g(x) = 4 – 3x h(x) = 2x – 1 (a) Find h(-2). … [1] (b) Find g-1(x). g-1(x) = … [2] (c) Find g(f(3)). … [2] (d) Find and simplify g(g(x)). … [2] (e) Find h-1(7). … [2] (f) Write as a single fraction in its simplest form. 1 1 + f x g x ^ h ^ h … [3]

12 marks

Mark scheme: 12(a) 3 1 – or – 0.75 4 12(b) 4 − x 2 M1 for x = 4 – 3y or y + 3x = 4 or oe final answer 3 y 4 y – 4 = –3x or = – x 3 3 12(c) –17 2 B1 for [f(3)] = 7 or M1 for 4 – 3(2x + 1) soi 12(d) 9x – 8 final answer 2 M1 for 4 – 3(4 – 3x) 12(e) 3 2 M1 for 2x – 1 = 7 or log2(x + 1) oe 12(f) 5 − x 3 M1 for 4 – 3x + 2x + 1 oe final answer M1 for common denominator (2 x + 1)(4 − 3x ) (2x + 1)(4 – 3x)

This question in 0607/43 May/June 2018

12 y 15 x –6 0 6 –10 (2x - 3) f (x) = (x + 2) (a) On the diagram, sketch the graph of y = f(x) for values of x between -6 and 6. [3] (b) Write down the equations of the asymptotes of y = f(x). … … [2] (c) g(x) = 5 - 2x (i) Solve f(x) = g(x). x = … or x = … [2] (ii) Find g(f(x)). Give your answer as a single fraction in its simplest form. … [3]

10 marks

Mark scheme: 12(a) Correct sketch 3 B1 for correct left hand branch without 15 y f(x)=(2x-3)/(x+2) serious curl back 10 5 B2 for correct right-hand branch x or B1 for correct shape right-hand branch but -6 6 with clear intercepts but serious overlap or -5 curl back -10 12(b) x = –2 oe 2 B1 for each y = 2 oe 12(c)(i) –2.81 or –2.812 to –2.811 2 B1 for each or for 2 x 2 + x − 13 = 0 2.31 or 2.311 to 2.312 12(c)(ii) x + 16 3 5( x + 2) − 2(2 x − 3) M2 for x + 2 x + 2  2 x − 3  or M1 for 5 −2  oe  x + 2 

This question in 0607/42 Oct/Nov 2018

10 (a) f ()x = 5 - 2x g ()x = 3x + 2 (i) Find f (- 3 ) . … [1] (ii) Find f (g (4)) . … [2] f (x) (iii) Solve = 2 . g (x) x = … [3] (iv) Find f - 1 ()x . f - 1 ()x = … [2] (v) Find and simplify g (f (x)) . … [2] (vi) Write as a single fraction in its simplest form. 3 2 + f (x) g ( x) … [3] (b) The function h ()x has an inverse function j ()x . Write down, in its simplest form, j (h (x)) . … [1]

14 marks

Mark scheme: 10(a)(i) 11 1 10(a)(ii) –23 2 M1 for 5 – 2(3 × 4 + 2) soi or 5 – 2(3x + 2) 10(a)(iii) 1 3 M1 for 5 – 2x = 2(3x + 2) oe oe M1FT for 5 – 4 = 6x + 2x or better 8 10(a)(iv) 5 −x 2 M1 for 2x + y = 5 or better oe final answer or x = 5 – 2y 2 y 5 or = − x 2 2 10(a)(v) 17 – 6x oe final answer 2 M1 for 3(5 – 2x) + 2 10(a)(vi) 5 x + 16 5 x + 16 3 M1 for common denominator or 2 (5 – 2x)(3x + 2) oe (5 − 2 x )(3 x + 2) 10 + 11x − 6 x M1 for 3(3x + 2) + 2(5 – 2x) oe final answer 10(b) x 1

This question in 0607/41 May/June 2019

12 f x = 10 - x g x = x 2 + 1 h x = j x = log 3 x x (a) Find g(3). … [1] (b) Find f(h(2)). … [2] (c) Find g(f(x)) in the form ax 2 + bx + c . … [3] (d) For some functions, p-1(x) = p(x). Write down which two functions, f(x), g(x), h(x) or j(x), have this property. … and … [2] 1 (e) Write h x - as a single fraction in its simplest form. ` j f x ` j … [3] (f) (i) Find j(243). … [1] (ii) Find x when j(x) = 1.5 . x = … [1] (iii) Find j-1(x). j – 1(x) = … [2]

15 marks

Mark scheme: 12(a) 10 1 12(b) 9.5 oe 2 1 1 M1 for 10 − soi e.g. 10 – x 2 12(c) x 2 − 20 x + 101 3 M1 for (10 −x ) 2 + 1 B1 for 100 – 10x – 10x + x2 oe 12(d) f(x) and h(x) 2 B1 for each 12(e) 10 − 2 x 3 M1 for common denominator x(10 – x) oe oe B1 for (10 – x) – x oe seen x (10 − x ) 12(f)(i) 5 1 12(f)(ii) 3 1 3 3 oe or 3 2 or 5.2[0] or 5.196... 12(f)(iii) 3x 2 M1 for x = log 3 y or x = 3 y

This question in 0607/42 May/June 2019

4 y 6 –270 0 270 x –6 (a) On the diagram, sketch the graph of y = f(x) where 1 f (x) = for values of x between -270 and 270 . cos x [3] (b) Write down the range of f(x). … [2] (c) (i) On the same diagram, sketch the graph of y = g(x) where (720 + x) g (x) = for values of x between -270 and 270 . [2] 2x (ii) Find the values of the x co-ordinates of the points of intersection of the two graphs. x = … or x = … or x = … [3] (iii) Find the equation of each asymptote of the graph of y = g(x). … [2]

12 marks

Mark scheme: 4(a) Correct sketch 3 B1 for correct shape B1 for max and min approx. correct B1 for asymptotes approx. correct 4(b) f( x ) - − 1 and f ( x ) . 1 2 B1 for each 4(c)(i) Correct sketch 2 B1 for each branch 4(c)(ii) –213 or –212.9 to –212.8 3 B1 for each –111 or –111.5 to –111.4 78.6[…] 4(c)(iii) x = 0 2 B1 for each y = 0.5

This question in 0607/43 May/June 2019

3 y 4 –3 0 3 x –2 1 f (x) = 3 x ! 1 (1 - x ), (a) On the diagram, sketch the graph of y = f ( x) for values of x between - 3 and 3. [3] (b) Write down the range of f(x) for - 3 G x G 0 . … [2] (c) On the same diagram, sketch the graph of y = x2 for - 2 G x G 2 . [1] 1 2 (d) (i) Solve the equation 3 = x . 1 - x x = … [1] 1 2 u w (ii) The equation 3 = x can be written in the form x - x + 1 = 0 . 1 - x Find the value of u and the value of w. u = … w = … [2]

9 marks

Mark scheme: 3(a) Correct sketch 3 B2 for first branch correct, including gradient 4444 zero at y-intercept or B1 for first branch above x-axis, increasing 3333 and crossing y-axis 2222 1111 B1 for second branch correct 3333 -2-2-2-2 -1-1-1-1 0000 0000 1111 2222 3333 -1-1-1-1 -2-2-2-2 3(b) 0.0357 or 0.03571... - f ( x ) - 1 oe 2 B1 for 0 < f ( x ) or f ( x ) - 1 3(c) Correct sketch 1 Vertex at origin 3(d)(i) –[0].809 or –[0].8087... 1 3(d)(ii) [u = ] 5 2 B1 for each [w = ] 2 or SC1 for answers reversed

This question in 0607/41 Oct/Nov 2019

10 f ()x = 2x + 3 g ()x = x , x ! 0 h ()x = 2 j (x ) = log 3 x x (a) Find (i) f (- 2), … [1] 1 (ii) . ge 2 o … [1] (b) Find g (f (1)). … [2] 1 (c) Find x when h (x) = . 8 x = … [1] (d) Find j (81 ) . … [1] (e) Find f (f (x)) in its simplest form. … [2] (f) Find f (x) # f (x) + f ( x) + 1 in its simplest form. … [3] (g) Find j -1 (x). j -1 ()x = … [2]

13 marks

Mark scheme: 10(a)(i) –1 1 10(a)(ii) 2 1 10(b) 1 2 B1 for 5 oe 5 1 1 or M1 for soi e.g. 2 x + 3 2(1) + 3 10(c) –3 1 10(d) 4 1 10(e) 4x + 9 2 M1 for 2(2x + 3) + 3 oe 10(f) 4 x 2 + 14 x + 13 3 M1 for (2 x + 3) 2 + 2 x + 3 + 1 oe B1 for [(2 x + 3) 2 = ] 4 x 2 + 12 x + 9 10(g) 3x 2 M1 for x = log 3 y or for x = 3y

This question in 0607/41 Oct/Nov 2019

2 f ()x = , x ! 2 g ()x = x + 2 h ()x = x 2 x - 2 (a) Find f (6 ). … [1] (b) Solve f (x) =- 2. x = … [2] (c) Find h (g (x)). … [1] (d) Solve h (g (x)) = h (x) + 2. x = … [4] (e) Find f - 1 (x). f - 1 ()x = … [3] (f) y 9 x –3 0 3 –3 (i) On the diagram, sketch the graph of y = f (x) and the graph of y = h (x) for values of x between - 3 and 3. [3] (ii) Write down the equation of the line of symmetry of y = h (x). … [1] (iii) Solve f (x) 2 h (x). … [2]

17 marks

Mark scheme: 2(a) 0.25 oe 1 2(b) 1.5 2 1 M1 for 1 = –2(x –2) or −= x – 22 2(c) (x + 2)2 1 2(d) 1 4 B1 for x 2 + 4 x + 4 –0.5 or − 2 M2 for 4 x + 4 = 2 or M1 for their ( c ) = x 2 + 2 or M2 for correct sketch or M1 for any U-shaped parabola 2(e) 1 3 1 1 + 2 oe final answer M2 for x = + 2 or xy = 1 + 2 y or y – 2 = x y x 1 or M1 for x − 2 = or y ( x − 2) = 1 y 1 or x = y − 2 2(f)(i) Correct sketches 3 B1 for correct quadratic shape through origin B2 for correct rectangular hyperbola shape or B1 for one branch 2(f)(ii) x = 0 1 2(f)(iii) 2 < x < 2.21 or 2.205 to 2.206 2 B1 for each part or 2 and 2.21 or 2.205 to 2.206 seen

This question in 0607/42 Oct/Nov 2019

13 f ()x = 2x + 5 g ()x = 1 - 2x (a) Find g (- 4) . … [1] (b) Find f - 1 (- 7) . … [2] (c) Find g (f (3)) . … [2] (d) Find and simplify f (g (x)) . … [2] (e) Find and simplify g -1 ()x . g -1 ()x = … [2] (f) Write as a single fraction, simplifying your answer. 3 2 + f ()x … [2]

11 marks

Mark scheme: 13(a) 9 1 13(b) –6 2 x− 5 M1for f(x) = –7 or for f–1(x) = 2 13(c) –21 2 B1 for 11 seen or M1 for 1 – 2(2x + 5) 13(d) 7 – 4x 2 M1 for 2(1 – 2x) + 5 13(e) 1 − x 2 M1 for 2x = 1 – y or x = 1 – 2y oe 2 y 1 or = – x 2 2 13(f) 4 x + 13 2 2(2 x + 5) + 3 final answer M1 for 2 x + 5 2 x + 5

This question in 0607/43 Oct/Nov 2019

10 f ( )x = 2 x + 3 g ( )x = 5 x (a) Find f ( g ( 3)) . … [2] (b) Find f -1 ( )x . f -1 ( )x = … [2] 1 (c) Find x when g ( )x = . 25 5 x = … [2] (d) Find g -1 ( )x . g -1 ( )x = … [2]

8 marks

This question in 0607/41 May/June 2020

9 f ( )x = 4 - 3 x g ( x) = , x ! 1 h ( )x = x 2 x - 1 (a) Find (i) f ( 2) , … [1] (ii) f ( g ( 4)) . … [2] (b) Find g ( g ( - 1)) . … [2] (c) Solve. h ( f ( x)) = 9 x = … or x = … [3] (d) Find ( f ( x)) 2 - 1 in terms of x. Give your answer in the form k ( ax + b)( cx + d) where a, b, c, d and k are integers. … [3]

11 marks

Mark scheme: 9(a)(i) –2 1 9(a)(ii) 3 2  1  M1 for f   oe  3  9(b) 2 2  1  − oe M1 for g  −  oe 3  2  9(c) 1 7 3 M2 for 4 − 3 x = ± 7 and or M1 for f(x) = ±3 3 3 If 0 scored B1 for each answer 9(d) 3(3 x − 5)( x − 1) 3 B2 for 9 x 2 − 12 x − 12 x + 15 or M1 for (4 − 3 x ) 2 − 1

This question in 0607/42 May/June 2020

12 f ( x) = 2x + 3 g ( )x = 5 - 3 x (a) Find f ( 4) . … [1] (b) Solve f ( x) - g ( x) = 5 . x = … [2] (c) Find g -1 ( )x . g -1 ( )x = … [2] (d) Find and simplify f ( g ( x)) . … [2] 2 3 (e) Simplify + . f ( x) g ( x) … [3]

10 marks

Mark scheme: 12(a) 11 1 12(b) 1.4 oe 2 M1 for 2x + 3x = 5 – 3 + 5 12(c) 5 − x 2 M1 for x = 5 – 3y or y – 5 = – 3x oe or oe 3 y 5 = – x oe 3 3 12(d) 13 – 6x 2 M1 for 2(5 – 3x) + 3 12(e) 19 3 M1 for 2(5 – 3x) + 3(2x + 3) final answer M1 for common denominator (2 x + 3)(5 − 3 x ) (2x + 3)(5 – 3x)

This question in 0607/43 May/June 2020

11 f ( x) = x3 g ( x) = 3x (a) Find g ( 2) - f ( 2) . … [2] 1 (b) Find x when g ( x) = . 9 … [1] 1 (c) Write x - in terms of x. f ( x) Give your answer as a single fraction. … [2] (d) Find f - 1 ( x) . f -1 ( x) = … [1]

6 marks

Mark scheme: 11(a) 1 2 B1 for 32 or 23 11(b) –2 1 11(c) x 4 − 1 2 1 final answer M1 for x − 3 3 x x 11(d) 3 x oe final answer 1

This question in 0607/43 Oct/Nov 2020

11 f ( )x = 3 x + 1 g ( )x = x 2 - 5 h ( )x = 3 x (a) Find g(3). … [1] (b) Find f(h(2)). … [2] (c) Find the value of r when f(r) = r. r = … [2] (d) Solve g(f(x)) = 20. x = … or x = … [3] (e) Find h -1 ( )x . h -1 ( )x = … [2]

10 marks

Mark scheme: 11(a) 4 1 11(b) 28 2 B1 for f(32) seen or M1 for 3 × 3x + 1 oe 11(c) 1 oe 2 M1 for 3r + 1 = r − 2 11(d) 4 3 M1 for (3x + 1)2 – 5 oe , −2 M1 for (3x + 1) = ±5 or [3](x + 2)(3x – 4) = 0 oe 3 or correct substitution in formula for 3x2 + 2x – 8 or 9x2 + 6x – 24 or correct and suitable sketch 11(e) log x 2 M1 for log y = log 3x oe or correct answer seen log 3 x or final answer log3 or x = 3y or log 3 y = x

This question in 0607/42 Feb/March 2021

5 f ( )x = 2 x - 1 g ( )x = 3 - x h ( )x = x 2 (a) Find (i) f ( - 2) , … [1] (ii) h ( g ( - 2)) . … [2] (b) Solve f ( )x = 7 . x = … [2] (c) Find f ( g ( x)) . … [1] (d) Solve f (x) # g (x) + 2h ( x) = 0 . x = … [3] (e) Find g -1 ( )x . g -1 ( )x = … [2] (f) y 10 – 3 0 3 x – 2 (i) On the diagram, sketch the graph of y = h ( x) for values of x between - 3 and 3. [2] (ii) Write down the equation of the line of symmetry of the graph of y = h ( x) . … [1] (iii) On the diagram, sketch the graph of y = g ( x) for values of x between - 3 and 3. [1] (iv) Solve g ( x) 2 h ( x) . … [2]

17 marks

Mark scheme: 5(a)(i) –5 1 5(a)(ii) 25 2 2 M1 for g(–2) = 5 or ( 3 − x ) soi 5(b) 4 2 M1 for 2x – 1 = 7 5(c) 5 – 2x oe Final answer 1 5(d) 3 3 2 2 oe B2 for 6 x −+3 x − 2 x + 2 x [= 0] or 7 better or B1 for 6 x −+3 x − 2 x 2 or M1 for ( 2 x − 1)( 3 − x ) + 2 x 2 [ = 0 ] 5(e) 3 − x Final answer 2 M1 for x = 3 – y or y + x = 3 5(f)(i) Correct sketch 2 B1 for any quadratic graph 5(f)(ii) x = 0 cao 1 5(f)(iii) Correct sketch 1 5(f)(iv) –2.3[0] < x < 1.3[0] 2 For x, do not allow f(x) or y for full or –2.303 to –2.302 < x < 1.302 to 1.303 marks B1 for either inequality or for 1.3[0] and –2.3[0] y range included scores 0

This question in 0607/41 May/June 2021

12 f ( )x = 2 - 3 x g ( )x = 2 - 3x (a) Find f(4). … [1] (b) Solve g(x) = 4. … [3] (c) Find f -1 ( )x . f -1 ( )x = … [2] (d) Find g ( f ( x)) . Write your answer as a single fraction in its simplest form. … [2] (e) Find f(x) - g(x). Write your answer as a single fraction in its simplest form. … [3]

11 marks

Mark scheme: 12(a) –10 1 12(b) 1 3 M2 for 5 = 8 – 12x oe oe 5 4 or M1 for = 4 2 − 3x 12(c) 2 − x 2 M1 for 3x + y = 2 or x = 2 – 3y oe y 2 3 or = − x or better 3 3 12(d) 5 2 5 oe final answer M1 for −+4 9x 2 − 3(2 − 3 )x 12(e) 9 x 2 − 12 x − 1 3 ( 2 − 3 x )( 2 − 3 x ) − 5 oe final answer M1 for 2 − 3 x 2 − 3 x B1 for 4 – 6x – 6x + 9x2

This question in 0607/42 May/June 2021

9 y 10 0 x 2.5 f ( x) = x x, x 2 0 (a) On the diagram, sketch the graph of y = f(x) for 0 1 x G 2.5 . [2] (b) Find the coordinates of the local minimum point. ( … , … ) [2] (c) (i) Find x when f(x) = 3x. … [3] (ii) Solve f ( )x H 3x . … [2]

9 marks

Mark scheme: 9(a) Correct sketch 2 B1 for correct shape but cutting either axis or without minimum 9(b) (0.368, 0.692) 2 B1 for each or (0.3678 to 0.3679, 0.6922...) 9(c)(i) 0.237 or 0.2369 to 0.2370 3 M1 for correct line sketched 2.31 or 2.311 … B1 for one correct 9(c)(ii) [0 <] x ⩽ 0.237 or 0.2369 to 2 B1 for each 0.2370 x ⩾ 2.31 or 2.311…

This question in 0607/43 May/June 2021

11 (a) f(x) = 3x + 2 g(x) = x2 h(x) = 2x (i) Find f(2). … [1] (ii) Find f(g(3)). … [2] h ( g ( 3)) (iii) Find the value of . g ( h ( 3)) … [3] (iv) Find f -1 ( )x . f -1 ( )x = … [2] (v) Find h -1 ( )x . h -1 ( )x = … [2] 1(b) (i) Find the value of log 3 81 - log 9 3 b l. … [2] 2 (ii) log b 25 = 3 Find the value of b. b = … [2]

14 marks

Mark scheme: 11(a)(i) 8 1 11(a)(ii) 29 2 M1 for g(3) = 9 or 3( x 2 ) + 2 11(a)(iii) 8 3 B1 for h(g(3)) = 29 or 512 B1 for g(h(3)) = 26 or 64 11(a)(iv) x − 2 2 y 2 oe final answer M1 for x = 3y – 2 or y − 2 = 3 x or = x + 3 3 3 11(a)(v) log x 2 M1 for x = 2y or x log2 = log y oe log 2 x or log 2 11(b)(i) 1 2   1  1 = 4 oe B1 for [log 3 81 = ]4 or  log 9    − 2   3   2 11(b)(ii) 125 2 2 B1 for 25 = b 3 oe

This question in 0607/43 May/June 2021

8 f ( x) = 2 x + 1 g ( x) = 3 - 2 x h ( x) = log ( x + 1) (a) Find the value of (i) f(12), … [1] (ii) g(f(12)). … [1] (b) Find the value of x when f ( x) = g ( x) . x = … [2] (c) Find f(g(x)), giving your answer in its simplest form. … [2] (d) Find g -1 ( x) . g -1 ( x) = … [2] (e) Find x when h (x) = f ( 0 .5 ) . x = … [2] (f) Find h -1 ( x) . h -1 ( x) = … [2]

12 marks

Mark scheme: 8(a)(i) 25 1 8(a)(ii) –47 1 FT 3 – 2 × their 25 8(b) 1 2 M1 for 2x + 1 = 3 – 2x or better oe 2 8(c) 7 – 4x final answer 2 M1 for 2(3 – 2x) + 1 8(d) 3 −x 2 M1 for y + 2x = 3 or better oe final answer y 3 2 or = − x 2 2 or x = 3 – 2y 8(e) 99 2 M1 for log( x + 1) = 2(0.5) + 1 or better 8(f) 10 x − 1 2 M1 for 10 y = x + 1 or x = log( y + 1)

This question in 0607/42 Feb/March 2022

3 y 5 – 3 0 1 x – 5 1 f ( x) = 2 x + 4 - 2 x (a) On the diagram, sketch the graph of y = f ( x) for values of x between - 3 and 1. [3] (b) Write down the equation of the asymptote of the graph. … [1] (c) Find the coordinates of the local maximum. ( … , … ) [1] (d) g( )x = x 3 - 5x for - 3 G x G 1. Solve f ( x) G g( x) . … [4]

9 marks

Mark scheme: 3(a) correct sketch 3 B2 for correct branches but joined or touching y-axis B1 for one correct branch 3(b) x = 0 1 3(c) (−1, 1) 1 3(d) −2.31 ⩽ x < 0 and 0 < x ⩽ 0.388 4 B3 for −2.3 ⩽ x ⩽ 0.388 or −2.31 ⩽ x ⩽ 0.39 or B2 for –2.3 ⩽ x ⩽ 0.39 or –2.31 ⩽ x or x ⩽ 0.388 or B1 for – 2.31 or 0.388 seen or for correct sketch

This question in 0607/41 May/June 2022

9 (a) f ( x) = 2 x + 3 g ( x) = 2 - 4 x h ( x) = 3x (i) Find f(5). … [1] (ii) Find and simplify g(f(x)). … [2] (iii) Find g -1 ( x) . g -1 ( x) = … [2] (iv) Solve h ( x) = 48 . … [2] (b) (i) The diagram shows a sketch of the graph of y = j ( x) . y 10 j(x) 5 – 10 – 5 0 5 10 x – 5 – 10 On the same diagram, sketch the graph of y = j ( x + 2) . [1] (ii) The diagram shows the graphs of y = k ( x) and y = m ( x) . y 10 k(x) m(x) 5 0 x – 5 3 – 5 – 10 Write k ( x) in terms of m ( x) . k ( x) = … [1]

9 marks

Mark scheme: 9(a)(i) 13 1 9(a)(ii) 8 x  10 oe final answer 2 M1 for 2  4(2 x  3) or better seen 9(a)(iii) 2 x 2 M1 for correct first step oe final answer 4 y 1 y  4 x  2 or   x or x  2  4 y 4 2 y – 2 = – 4x 9(a)(iv) 3.52 or 3.523 to 3.524 or log 3 48 or 2 M1 for log48  log3 x or better log48 or suitable sketch log3 9(b)(i) 1 9(b)(ii) 2m(x) 1

This question in 0607/41 May/June 2022

9 f ( x) = 2 - 3 x g ( x) = ( x + 1) 2 h ( x) = log x (a) Find. (i) f ( - 4) … [1] (ii) f ( g ( 3)) … [2] (iii) f -1 ( 4) … [2] (iv) h -1 ( 2) … [2] (b) Solve ( f ( x)) -1 = 5 . x = … [3] (c) Find g ( f ( x)) . Write your answer in the form ax 2 + bx + c . … [3] (d) y = h ( f ( x)) Find x in terms of y. x = … [3]

16 marks

Mark scheme: 9(a)(i) 14 1 9(a)(ii) –46 2 B1 for f(16) or M1 for 2 – 3(x + 1)2 soi 9(a)(iii) 2 2 2  x  oe M1 for 2 – 3x = 4 or 3 3 9(a)(iv) 100 2 M1 for logx = 2 or 10x 9(b) 3 3 M2 for 1 = 5(2 – 3x) oe oe 5 1 or M1 for [=5] 2  3x 9(c) 9x2 – 18x + 9 3 M1 for (2 – 3x + 1)2 M1FT for 32 – 3 × 3x – 3 × 3x + (3x)2 9(d) 2  10 y 3 M2 for 10y = 2 – 3x oe or M1 for y orh( x )  log(2  3 x ) 3 or for 10y seen

This question in 0607/43 May/June 2022

7 (a) Solve the simultaneous equations. You must show all your working. 4x + 3y = - 21 6x - 2y = 1 x = … y = … [4] 1 3(b) f ( x) = 5 x - 2 g ( x) = , x ! 0.5 h ( x) = ( x - 1 ) 2x - 1 (i) Find f ( 3) . … [1] (ii) Find h ( f ( 2)) . … [2] (iii) Solve f ( h ( x)) = - 7 . x = … [3] (iv) Find g ( g ( x)) in terms of x. Give your answer in its simplest form. … [3]

13 marks

Mark scheme: 7(a) Correctly equating coefficients M1 or correctly isolating x or y Correct method to eliminate one variable M1 [x = ] –1.5 A1 [y = ] –5 A1 If second M is M0, SC1 for answers that satisfy one equation or if 2 correct answers and no working shown 7(b)(i) 13 1 7(b)(ii) 343 2 M1 for ((5 x − 2) − 1) 3 or ((5(2) − 2) − 1) 3 or for h(8) used correctly 7(b)(iii) 0 3 B2 for ( x − 1) 3 = − 1 or M1 for 5( x − 1) 3 − 2 [ = − 7] oe 7(b)(iv) 2 x − 1 3 1 oe final answer B2 for or better 3 − 2 x  2 − 2 x + 1     2 x − 1  1 or M1 for  1 2  − 1  2 x − 1  8 In parts (a), (b) and (c), marks can only be earned with an increasing curve or plots

This question in 0607/41 Oct/Nov 2022

11 y 8 – 5 0 5 x – 9 5 f ( x) = x + ( x - 2)( x + 3) (a) Sketch the graph of y = f ( x) for values of x between - 5 and 5. [4] (b) Write down the equations of the asymptotes parallel to the y-axis. … [2] (c) (i) Find the coordinates of the local maximum. ( … , … ) [2] (ii) Find the coordinates of the local minimum. ( … , … ) [2] (iii) Write down the range of values of k for which f ( x) = k has exactly one solution. … [2] (d) g ( x) =- 4 - x (i) Solve the equation f ( x) = g ( x) . … [3] (ii) Find the solutions to the inequality f ( x) 2 g ( x) . … [3] Question 12 is printed on the next page.

18 marks

Mark scheme: 11(a) Correct sketch 4 B1 for each outside branch B2 for middle branch with offset maximum or B1 if not offset or if offset crosses x- axis 11(b) x = 2, x = –3 2 B1 for each 11(c)(i) (1.03, –0.249) 2 1.029... –0.2491 to –0.2490 B1 for each coordinate 11(c)(ii) (2.99, 3.83) 2 2.986... 3.833... B1 for each coordinate 11(c)(iii) –0.249 < k < 3.83 2 B1FT for each 11(d)(i) –3.35 or –3.347..., –1.52 or –1.520... 3 B1 for each 1.87 or 1.867... If 0 scored, SC1 for y = –4 – x sketched on diagram or for –3.3, –1.5, and 1.9 or if y-coordinates also given 11(d)(ii) –3.35 < x < –3, 3 B1FT for each –1.52 < x < 1.87, x > 2

This question in 0607/42 Oct/Nov 2022

2 y 1 – 5 0 5 x – 3 1 1 f ( x) = - 2 x x (a) On the diagram, sketch the graph of y = f ( x) for values of x between - 5 and 5. [2] (b) Find f ( - 2) . … [1] (c) Solve the equation f ( x) = 0 . x = … [1] (d) Find the maximum value of f(x). … [1] (e) Write down the equation of each asymptote. … [2] (f) (i) Solve the equation. 1 1 2 - = x - 2 2 x x … [3] 1 1 2 4 2 (ii) The equation - 2 = x - 2 can be rearranged to the form x + ax + bx + c = 0 . x x Find the values of a, b and c. a = … b = … c = … [2]

12 marks

Mark scheme: 2(a) Correct sketch 2 No intersections with y-axis B1 for each branch with no large curl back or feathering. Right hand branch with a maximum or level. 2(b) –[0].75 oe 1 2(c) 1 1 2(d) 0.25 oe 1 Not coordinates 2(e) x = 0 2 B1 for each y = 0 2(f)(i) 0.525 or 0.5248 to 0.5249 3 B2 for one correct or M1 for sketch of y = x2 – 2 added 1.49 or 1.490... to diagram 2(f)(ii) [a =] –2 2 B1 for x −=1 x 4 − 2 x 2 oe [b =] –1 [c =] 1

This question in 0607/43 Oct/Nov 2022

9 (a) f ( x) = 2 x + 3 g ( x) = x 2 + 1 h ( x) = 2 sin ( 2x) (i) Find f ( - 2) . … [1] (ii) Find f -1 ( x) . f -1 ( x) = … [2] (iii) Find x when g ( x) = 2 f ( x) . x = … or x = … [3] (iv) Find g ( f ( x)) , giving your answer in the form ax 2 + bx + c . … [3] (v) Find the amplitude and period of h ( x) . Amplitude = … Period = … [2] (vi) Solve the equation h ( x) = 3 for 0° G x G 180 ° . … [2] (b) j ( x) = log a x, x 2 0 (i) Find the value of j 3 a ` j. … [1] (ii) Find j -1 ( x) . j -1 ( x) = … [2]

16 marks

Mark scheme: 9(a)(i) –1 1 9(a)(ii) x − 3 2 y 3 oe final answer M1 for y – 3 = 2x or = x + 2 2 2 or x = 2y + 3 9(a)(iii) –1, 5 3 B2 for (x – 5)(x + 1) or sketch indicating –1 and 5 −−( 4 )  ( −4 ) 2 − 4 (1)( −5 ) or oe 2 (1) or M1 for x 2 + 1 = 2(2 x + 3) oe 9(a)(iv) 4x2 + 12x + 10 cao 3 M1 for (2x + 3)2 + 1 B1 for  ( 2 x + 3 ) 2 =  4 x 2 + 6 x + 6 x + 9   or 4 x 2 + 12 x + 9 9(a)(v) 2 2 B1 for each 180 9(a)(vi) 30, 60 2 B1 for each 9(b)(i) 1 1 oe 3 9(b)(ii) ax final answer 2 M1 for a y = x or x = loga y

This question in 0607/43 Oct/Nov 2022

6 f ( )x = 3 - 2 x g ( )x = x + 1 h ( x) = ( x + 1) 2 j ( x) = tan x° for 0 1 x 1 180 (a) Find f ( - 1.5) . … [1] (b) Find h(h(2)). … [2] (c) Find g(f(x)), giving your answer in its simplest form. … [2] (d) Find f -1 ( )x . f -1 ( )x = … [2] (e) Find x when j -1 ( )x = 75 . … [2]

9 marks

Mark scheme: 6(a) 6 1 6(b) 100 2 M1 for h((2  1)2) oe or ((x  1)2  1)2 6(c) 4 – 2x or 2(2 – x) final answer 2 M1 for 3 – 2x + 1 6(d) 3 x 2 y 3 oe final answer M1 for y  2x = 3 or for  – x or 2 2 2 3 y for x 3 – 2y or y – 3 = –2x or 2 oe 6(e) 3.73 or 3.732… or 2  3 2 M1 j(75) oe

This question in 0607/41 May/June 2023

11 f ( x) = 2 x + 5 g ( x) = 1 - 3 x (a) Find f ( - 2) . … [1] (b) Solve f ( g ( x)) = 19 . … [3] (c) Find g -1 ( x) . g -1 ( x) = … [2] g ( x) (d) y = f ( x) Find x in terms of y. x = … [3]

9 marks

Mark scheme: 11(a) 1 1 11(b) –2 3 B2 for –6x = 12 oe or better or M1 for 2(1 – 3x) + 5 = 19 11(c) 1  x 2 y 1 oe Final answer M1 for x = 1 – 3y or y + 3x = 1 or   x 3 3 3 or y – 1 = –3x 11(d) 1  5 y 3 M1 for y(2x + 5) = 1 – 3x oe oe Final answer M1FT dep for 2xy + 3x = 1 – 5y dependent on 2 y  3 4 term equation with 2 terms in x. M1FT for factorising and dividing to form a  by c  dy Max 2 marks if final answer is incorrect.

This question in 0607/42 May/June 2023

8 (a) Solve the simultaneous equations. 5x - 4y = 13 3x + 2y =- 1 You must show all your working. x = … y = … [3] 1 (b) f ( x) = 3 x + 1 g ( x) = , x ! 1.5 2x - 3 (i) Find f ( - 2) . … [1] (ii) Find f ( f ( x)) , giving your answer in its simplest form. … [2] 1 (iii) Solve g ( f ( x)) = . 5 x = … [3]

9 marks

Mark scheme: 8(a) Correct method to eliminate one variable M1 [ x  ] 1 A2 If 0 scored, SC1 for answers that [ y ] 2 satisfy one equation 8(b)(i) –5 1 8(b)(ii) 9x + 4 final answer M1 for 3  3 x  1  1 oe 2 8(b)(iii) 1 3 M2 for 2  3 x  1  3  5 oe 1 1 or M1 for  oe 2  3 x  1  3 5 OR M2 for f(x) = 4 oe 1 1 or M1 for  2f x  3 5

This question in 0607/43 May/June 2023

5 f ( x) = 2 x - 5 g ( x) = x 2 + x + 3 h ( x) = x3 j ( x) = 3x (a) The domain of f ( x) is 0 G x G 10 . Find the range of f ( x) . … [2] (b) Solve. (i) f ( x) =- 2 x = … [2] (ii) g ( x) = 3 - x x = … or x = … [3] (c) Find g ( f ( 4)) . … [2] (d) Find h ( 2) - j ( 2) . … [2] (e) Find h -1 ( x) . h -1 ( x) = … [1] (f) Find j -1 ( x) . j -1 ( x) = … [2]

14 marks

Mark scheme: 5(a) –5 ⩽ f(x) ⩽ 15 2 B1 for each If 0 scored, SC1 for -5 and 15 seen 5(b)(i) 1.5 oe 2 M1 for 2x = –2 + 5 5(b)(ii) –2 and 0 3 B2 for x2 + 2x = 0 oe or M1 for x2 + x +3 = 3 – x 5(c) 15 2 B1 for f(4) = 3 stated or used twice or M1 for (2x – 5)2 + (2x–5) + 3 oe 5(d) –1 2 M1 for 23 – 32 oe 5(e) 3 x oe 1 5(f) log x 2 log y log 3 x or M1 for x = log 3 y or x = log3 log3 or for x = 3y

This question in 0607/41 Oct/Nov 2023

2 f ( x) = 2 x + 4 g ( x) = x - 1 h ( )x = x 2 - 3x (a) Find (i) f ( 3) … [1] (ii) h ( 7) . … [1] (b) Find the value of x when g ( x) =- 6 . x = … [1] (c) Find f -1 ( x) . f -1 ( x) = … [2] (d) Simplify f ( x) # g ( x) + 1. … [2] (e) Solve h ( g ( x)) = 0 . x = … or … [3]

10 marks

Mark scheme: 2(a)(i) 10 1 2(a)(ii) 28 1 2(b) −5 1 2(c) x − 4 2 y 4 oe M1 for 2 x = y − 4 or x = 2 y + 4 or = x + 2 2 2 2(d) 2 x 2 + 2 x − 3 2 M1 for 2 x 2 + 4 x − 2 x − 4 [+1] 2 − 3( x − 1) or better 2(e)  x = 1 and  x =  4 3 M1 for ( x − 1) M1 for ( x − 1)( x − 4) = 0 or dep M1 for correct use of formula on their quadratic equation or dep M1 for sketch of their quadratic equation clearly showing 2 intersections with x-axis

This question in 0607/42 Oct/Nov 2023

10 f ( x) = 4x - 1 g ( x) = 3 - 2x h ( x) = 4 ( 2 - x) (a) (i) Find g ( -3) . … [1] (ii) Find f(h(4)). … [2] (iii) Find g ( f ( x)) . Give your answer in its simplest form. … [2] (iv) Find h -1 ( x) . h -1 ( x) = … [2] f ( x) (b) (i) Sketch the graph of y = for values of x between -2 and 4. g ( x) y 10 0 x -2 4 -10 [3] (ii) Write down the equation of the asymptote which is parallel to the y-axis. … [1] f ( x) (iii) Use the graph to solve h ( x) = . g ( x) x = … or x = … [3] f ( x) 2 (iv) h ( x) = can be rearranged to the form ax + bx + c = 0 . g ( x) Find the value of a, the value of b and the value of c. a = … b = … c = … [3]

17 marks

Mark scheme: 10(a)(i) 9 1 10(a)(ii) –33 2 B1 for [h(4)] = –8 or M1 for 4 × 4(2 – 4) – 1 10(a)(iii) 5 – 8x final answer 2 M1 for 3 – 2(4x – 1) or better 10(a)(iv) x 2 y 2 – oe final answer M1 for x = 4(2 – y) or = 2 − x 4 4 or y – 8 = – 4x 10(b)(i) Correct sketch 3 B2 for correct left-hand branch or B1 for left-hand branch with a positive y-intercept or passing through origin AND B1 for correct right-hand branch 10(b)(ii) x = 1.5 oe 1 10(b)(iii) 1.06 or 1.064 to 1.065 3 B2 for one correct 2.94 or 2.935... or M1 for sketch of y = 4(2 – x) 10(b)(iv) 8, –32, 25 3 B2 for 2 correct OR M1 for 4 ( 2 − x )( 3 − 2 x ) = 4 x − 1 B1 for 6 – 7x + 2x2 or 24 – 28x + 8x2

This question in 0607/43 Oct/Nov 2023

10 f ( x) = 3 x - 2 g ( x) = 5 - 2 x h ( x) = x2 (a) (i) Find g ( - 2) . … [1] (ii) Find h ( g ( x)) . Write your answer in the form ax 2 + bx + c . … [3] (iii) Find g -1 ( x) . g -1 ( x) = … [2] f ( x) (b) (i) On the diagram, sketch the graph of y = for values of x between -2 and 6 . g ( x) y 10 0 x -2 6 -10 [3] f ( x) (ii) An asymptote to the graph of y = is parallel to the y-axis. g ( x) Find the equation of this asymptote. … [1] f (x) (iii) Solve = 5 - 2x . g (x) … [3]

13 marks

Mark scheme: 10(a)(i) 9 1 10(a)(ii) 4x2 – 20x + 25 final answer 3 M1 for (5 – 2x)2 B1 for 3 terms correct in 25 – 10x – 10x + 4x2 10(a)(iii) 5 −x 2 y 5 oe final answer M1 for x = 5 – 2y or = − x 2 2 2 or y – 5 = –2x 10(b)(i) Correct graph 3 B2 for correct but some ‘curl back’ or overlap or too wide a gap. B1 for one branch correct. 10(b)(ii) x = 2.5 oe 1 10(b)(iii) 1.67 or 1.670 to 1.671 3 B2 for one solution 3.31 or 3.307 to 3.308 Max 1 if y coordinates included. or M1 for sketch of y = 5 – 2x If 0 scored, SC1 for only y values seen in answer space, with correct x values seen in working.

This question in 0607/42 Feb/March 2024

11 f( )x = 3 x - 1 g( )x = 5 - 2 x h( x) = , x ! 1 .5 2x - 3 (a) Find f( 4) . … [1] (b) Solve f( )x = - 7 . … [2] (c) Find g -1 ( )x . g -1 ( )x = … [2] (d) Solve g( x) = 7 h(f( x)) . You must show all your working. x = … [6]

11 marks

Mark scheme: 11(a) 11 1 11(b) –2 2 M1 for 3x = – 7 + 1 11(c) 5 x 2 y 5 oe final answer M1 for x = 5 – 2y or   x or 2x = 5 – y 2 2 2 11(d) 1 M1  h(f ( x ))   2(3 x  1)  3 (5  2 x )(6 x  5)  7 A1 All further FTs dep on second stage in correct form (5 – 2x)(ax + b) = k where a, and b are integers or sketch of rectangular hyperbola Correct expansion of brackets M1 30x – 25 – 12x2 + 10x [= 7] or sketch of straight line with negative gradient Correct rearrangement to 3 term quadratic M1 12 x 2  40 x  32  0 oe on one side or graphs intersecting twice in 1st quadrant Correct factorisation M1 (6 x  8)(2 x  4)  0 oe or correct use of formula or correct sketch of the quadratic or solutions indicated at points of intersection 4 B1 Both answers correct 2, oe 3

This question in 0607/41 May/June 2024

9 (a) f ( )x = 3 + 2 x g ( )x = x 2 + 1 h ( )x = x 5 (i) Find f ( - 5 ) . … [1] (ii) Find the value of h(f(9)). Give your answer in standard form correct to 4 significant figures. … [3] (iii) Find g(f(x)), giving your answer in the form ax 2 + bx + c . … [3] (iv) Find f - 1 ( )x . f - 1 ( )x = … [2] (v) The domain of h(x) is - 1 G x G 2 . Find the range of h(x). … [2] (b) j (x) = log (2x), x 2 0 (i) Find x when j ( )x = 3 . … [2] (ii) Find j -1 ( )x . j -1 ( )x = … [2] (iii) j ( w) = 3j ( x) Find w in terms of x. w = … [2]

17 marks

Mark scheme: 9(a)(i) –7 1 9(a)(ii) 4.084  10 6 cao 3 B2 for 4 084 101 or 4.084 101 × 106 or 4.0841[0] × 106 or answer 4.08 × 106 or M1 for (3  2  9) 5 If 0 scored, SC1 for their 5 or more figure answer in standard form and corrected to 4 sf. or for 4 084 000 seen 9(a)(iii) 4 x 2  12 x  10 final answer 3 M1 for (3  2 x ) 2  1 B1 for [(3  2 x ) 2  ] 9  6 x  6 x  4 x 2 9(a)(iv) x  3 2 y 3 oe final answer M1 for x  3  2 y or y  3  2 x or   x 2 2 2 9(a)(v) –1 ⩽ h(x) ⩽ 32 2 B1 for –1 ⩽ h(x) ⩽ k or k ⩽ h(x) ⩽ 32 or –1 and 32 evaluated 9(b)(i) 500 2 M1 for 2 x  10 3 9(b)(ii) 10 x 2 M1 for 2 x  10 y or x  log(2 y ) oe final answer 2 9(b)(iii) 4x3 final answer 2 M1 for [3log(2x) =] log(2x)3 oe 10 For all parts accept decimals or percentages with the usual rules for 3sf Do not penalise incorrect cancelling or converting Do not accept ratios or words

This question in 0607/42 May/June 2024

10 f ( x) = 5 - x g ( x) = 3 ( x + 1) h ( x) = sin xc for 0 G x G 180 2 (a) Find f ( 3) . … [1] (b) Solve f ( x) = 2 . x = … [2] (c) Find and simplify f ( g ( x) ) . … [2] (d) Find g -1( )x . g -1( )x = … [2] (e) Find h ( g ( 29)) . … [2] (f) Using a graphical method, solve h ( g ( x)) = 1 - 0. 01x . y 2 0 x 180 – 2 … [5]

14 marks

Mark scheme: 10(a) 1 1 3 oe 2 10(b) 6 2 1 M1 for 5 – x = 2 2 10(c) 1 1 7 x3 2 1 3  1 x or oe Final answer M1 for 5  (3( x  1)) oe 2 2 2 2 10(d) x  3 2 y oe Final answer M1 for x = 3(y + 1) or x + 1 = or y – 3 = 3x 3 3 10(e) 1 2 B1 for h(90) or M1 for sin(3(x + 1)) oe 10(f) sin(3(x + 1)) soi 1 Correct sketches e.g. 2 or a single graph of h(g(x)) – 1 + 0.01x B1 for each graph 17.5 or 17.52... 2 B1 for 1 correct. 48.7 or 48.71... 115.9 or 115.94...

This question in 0607/43 May/June 2024

6 f ( )x = 5x - 1 g ( )x = x 2 + x h ( x) = ( x - 1) 3 The domain for all three functions is x 2 2 . (a) Find f ( 3 ) . … [1] (b) Find the range of f ( )x . … [1] (c) Find g ( f ( 4)) . … [2] (d) Find h -1 ( )x . h -1 ( )x = … [2] (e) Simplify fully. 10h ( x) f ( x) - 4 … [3]

9 marks

Mark scheme: 6(a) 14 1 6(b) f(x) > 9 1 6(c) 380 2 M1 for g(19) or better or (5 x − 1) 2 + (5 x − 1) 3 y = x − 1 or x = ( y − 1)36(d) 1+ 3 x oe 2 M1 for 6(e) 2( x − 1) 2 nfww 3 10( x − 1) 3 M2 for or better seen 5( x − 1) Or M1 for 5 x −−1 4 or better seen

This question in 0607/43 Oct/Nov 2024

13 f ( )x = 4x - 2 g ( x) = ( x + 1 ) 2 (a) Find f ( 3 ) . … [1] (b) Find fg ( )x . Simplify your answer. … [2] (c) Find f -1 ( )x . f -1 ( )x = … [2] (d) Find ff -1 ( 5 ) . … [1]

6 marks

Mark scheme: 13(a) 10 1 13(b) 4x2 + 8x + 2 or 2(2x2 + 4x + 1) 2 M1 for 4(x + 1)2 – 2 or better final answer 13(c) x + 2 2 y 2 oe final answer M1 for x = 4y – 2 or y + 2 = 4x or = x − 4 4 4 13(d) 5 cao 1

This question in 0607/42 Feb/March 2025

17 f ( )x = 3 - 2x g ( )x = 1 - 5x (a) Find f ( - 2 ) . … [1] (b) Find x when f ( )x = 6 . … [2] (c) Find fg ( )x . Give your answer in its simplest form. … [2] (d) Find g -1 ( )x . g -1 ( )x = … [2]

7 marks

Mark scheme: 17(a) 7 1 17(b) 3 1 2 M1 for –2x = 6 – 3 oe or better – or − or –1.5 2 12 17(c) 1 + 10x final answer 2 M1 for 3 – 2(1 – 5x) 17(d) 1 −x 2 y 1 oe final answer M1 for x = 1 – 5y or 5x = 1 – y or = − x 5 5 5 or y – 1 = –5x or better

This question in 0607/43 May/June 2025